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<h1 class="title toc-ignore">Signal modeling</h1>
<h4 class="date">Last updated: May 07, 2024</h4>
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<p><br></p>
<p>This page describes the functions Gannet uses to model metabolite
signals. Note that when the definition of a parameter is omitted from a
table under a particular metabolite, it is implied that it has been
defined already in a previously described function.</p>
<p>For all model fitting, Gannet uses nonlinear regression, with fit
parameters optimized using the least-squares Levenberg-Marquardt
algorithm. For increased computational speed and a better solution, the
starting values of the optimization are derived from a “pre-fit” that
uses the trust-region-reflective algorithm. Description of these
algorithms can be found in the
<a href="https://www.mathworks.com/help/optim/ug/least-squares-model-fitting-algorithms.html" target="_blank">online
MATLAB documentation</a>.</p>
<div id="gabaglx" class="section level2 hasAnchor">
<h2 class="hasAnchor">GABA+Glx<a href="#gabaglx" class="anchor-section"
aria-label="Anchor link to header"></a></h2>
<p>GABA and Glx are fitted using a three-Gaussian model with a linear
slope and non-linear baseline:</p>
<p><span class="math display">\[
S(f) =
\sum_{i=1}^{3}\left\{A_i\exp[\sigma_i(f-f_i)^2]\right\}+
m(f-f_1)+
b_1\sin(\pi{f}/1.31/4)+
b_2\cos(\pi{f}/1.31/4)
\]</span></p>
<p>where:</p>
<table>
<thead>
<tr class="header">
<th align="left"><u>Parameter</u></th>
<th align="left"><u>Definition</u></th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td align="left"><span class="math inline">\(f\)</span></td>
<td align="left">Frequency (ppm)</td>
</tr>
<tr class="even">
<td align="left"><span class="math inline">\(A_i\)</span></td>
<td align="left">Gaussian <em>i</em>’s amplitude</td>
</tr>
<tr class="odd">
<td align="left"><span class="math inline">\(\sigma_i\)</span></td>
<td align="left">Gaussian <em>i</em>’s width</td>
</tr>
<tr class="even">
<td align="left"><span class="math inline">\(f_i\)</span></td>
<td align="left">Gaussian <em>i</em>’s center frequency (ppm)</td>
</tr>
<tr class="odd">
<td align="left"><span class="math inline">\(m\)</span></td>
<td align="left">Slope of linear baseline</td>
</tr>
<tr class="even">
<td align="left"><span class="math inline">\(b_1\)</span></td>
<td align="left">Sine baseline term</td>
</tr>
<tr class="odd">
<td align="left"><span class="math inline">\(b_2\)</span></td>
<td align="left">Cosine baseline term</td>
</tr>
</tbody>
</table>
<div class="info">
<p><i class="fa fa-info-circle" style="color: white"></i> The GABA+Glx
model is fitted using a model that has observation weights between 3.16
and 3.285 ppm, where the Cho subtraction artifact<span
class="citation"><sup><a href="#ref-Evans2013">1</a></sup></span>
appears. The purpose is to down-weight the influence of this artifact
(if present) on the model fitting.</p>
</div>
<p><img id="img_75" src="images/signal-modeling/GABA+Glx.png" alt="Illustration of the GABA+Glx model"></p>
</div>
<div id="gsh-te-100-ms" class="section level2 hasAnchor">
<h2 class="hasAnchor">GSH (TE < 100 ms)<a href="#gsh-te-100-ms"
class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>GSH that is edited at a TE < 100 ms is fitted with a five-Gaussian
model with a linear + quadratic baseline:</p>
<p><span class="math display">\[
S(f) =
\sum_{i=1}^{5}\left\{A_i\exp[\sigma_i(f-f_i)^2]\right\}+
m_1(f-f_1)+
m_2(f-f_1)^2+b
\]</span></p>
<p>where:</p>
<table>
<thead>
<tr class="header">
<th align="left"><u>Parameter</u></th>
<th align="left"><u>Definition</u></th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td align="left"><span class="math inline">\(m_1\)</span></td>
<td align="left">Slope of linear baseline</td>
</tr>
<tr class="even">
<td align="left"><span class="math inline">\(m_2\)</span></td>
<td align="left">Quadratic baseline term</td>
</tr>
<tr class="odd">
<td align="left"><span class="math inline">\(b\)</span></td>
<td align="left">Baseline offset</td>
</tr>
</tbody>
</table>
<p><img id="img_75" src="images/signal-modeling/GSH-80.png" alt="Illustration of the GSH model at TE = 80 ms"></p>
</div>
<div id="gsh-te-100-ms-1" class="section level2 hasAnchor">
<h2 class="hasAnchor">GSH (TE >= 100 ms)<a href="#gsh-te-100-ms-1"
class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>GSH that is edited at a TE >= 100 ms is fitted with a six-Gaussian
model with a linear + quadratic baseline:</p>
<p><span class="math display">\[
S(f) =
\sum_{i=1}^{6}\left\{A_i\exp[\sigma_i(f-f_i)^2]\right\}+
m_1(f-f_1)+
m_2(f-f_1)^2+b
\]</span></p>
<p><img id="img_75" src="images/signal-modeling/GSH-120.png" alt="Illustration of the GSH model at TE = 120 ms"></p>
</div>
<div id="lac" class="section level2 hasAnchor">
<h2 class="hasAnchor">Lac<a href="#lac" class="anchor-section"
aria-label="Anchor link to header"></a></h2>
<div class="info">
<p><i class="fa fa-info-circle" style="color: white"></i> Optimization
of the modeling of edited Lac is ongoing.</p>
</div>
<p>Lac is fitted with a four-Gaussian model with a linear + quadratic
baseline:</p>
<p><span class="math display">\[
S(f) =
\sum_{i=1}^{4}\left\{A_i\exp[\sigma_i(f-f_i)^2]\right\}+
m_1(f-f_1)+
m_2(f-f_1)^2+b
\]</span></p>
<p><img id="img_75" src="images/signal-modeling/Lac.png" alt="Illustration of the Lac model"></p>
</div>
<div id="etoh" class="section level2 hasAnchor">
<h2 class="hasAnchor">EtOH<a href="#etoh" class="anchor-section"
aria-label="Anchor link to header"></a></h2>
<p>EtOH is fitted with a two-Lorentzian model with a linear
baseline:</p>
<p><span class="math display">\[
S(f) =
\sum_{i=1}^{2}\left[\frac{A_{i}}{1+\left(\frac{f-f_{i}}{\gamma_{i}/2}\right)^2}\right]+
m(f-f_1)+b
\]</span></p>
<p>where:</p>
<table>
<thead>
<tr class="header">
<th align="left"><u>Parameter</u></th>
<th align="left"><u>Definition</u></th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td align="left"><span class="math inline">\(A_i\)</span></td>
<td align="left">Lorentzian <em>i</em>’s amplitude</td>
</tr>
<tr class="even">
<td align="left"><span class="math inline">\(f_i\)</span></td>
<td align="left">Lorentzian <em>i</em>’s center frequency (ppm)</td>
</tr>
<tr class="odd">
<td align="left"><span class="math inline">\(\gamma\)</span></td>
<td align="left">Lorentzian width (full-width at half-maximum)</td>
</tr>
</tbody>
</table>
<div class="info">
<p><i class="fa fa-info-circle" style="color: white"></i> The EtOH
model is fitted using a model that has observation weights between 1.29
and 1.51 ppm, where the Lac subtraction artifact appears. The purpose is
to down-weight the influence of this artifact (if present) on the model
fitting.</p>
</div>
</div>
<div id="chocr" class="section level2 hasAnchor">
<h2 class="hasAnchor">Cho+Cr<a href="#chocr" class="anchor-section"
aria-label="Anchor link to header"></a></h2>
<p>Cho and Cr in the edit-OFF spectrum are fitted with a two-Lorentzian
model with a linear baseline:</p>
<p><span class="math display">\[
Absorption(f) =
\frac{A}{2\pi}\frac{\gamma}{(f-f_0)^2+\gamma^2}+
\frac{Ah}{2\pi}\frac{\gamma}{(f-f_0-0.18)^2+\gamma^2}
\]</span> <span class="math display">\[
Dispersion(f) =
\frac{A}{2\pi}\frac{f-f_0}{(f-f_0)^2+\gamma^2}+
\frac{Ah}{2\pi}\frac{f-f_0-0.18}{(f-f_0-0.18)^2+\gamma^2}
\]</span></p>
<p><span class="math display">\[
S(f) =
\cos(\phi)Absorption(f)+
\sin(\phi)Dispersion(f)+
m(f-f_0)+b
\]</span></p>
<p>where:</p>
<table>
<thead>
<tr class="header">
<th align="left"><u>Parameter</u></th>
<th align="left"><u>Definition</u></th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td align="left"><span class="math inline">\(A\)</span></td>
<td align="left">Amplitude of Cr peak</td>
</tr>
<tr class="even">
<td align="left"><span class="math inline">\(\gamma\)</span></td>
<td align="left">Lorentzian width (half-width at half-maximum)</td>
</tr>
<tr class="odd">
<td align="left"><span class="math inline">\(f_0\)</span></td>
<td align="left">Center frequency of Cr peak</td>
</tr>
<tr class="even">
<td align="left"><span class="math inline">\(h\)</span></td>
<td align="left">Amplitude scaling factor for Cho peak</td>
</tr>
<tr class="odd">
<td align="left"><span class="math inline">\(\phi\)</span></td>
<td align="left">Phase</td>
</tr>
</tbody>
</table>
<p><img id="img_75" src="images/signal-modeling/Cho+Cr.png" alt="Illustration of the Cho+Cr model"></p>
</div>
<div id="naa" class="section level2 hasAnchor">
<h2 class="hasAnchor">NAA<a href="#naa" class="anchor-section"
aria-label="Anchor link to header"></a></h2>
<p>NAA in the edit-OFF spectrum is fitted with a Lorentzian model with a
linear baseline:</p>
<p><span class="math display">\[
Absorption(f) =
\frac{A}{2\pi}\frac{\gamma}{(f-f_0)^2+\gamma^2}
\]</span> <span class="math display">\[
Dispersion(f) =
\frac{A}{2\pi}\frac{(f-f_0)}{(f-f_0)^2+\gamma^2}
\]</span></p>
<p><span class="math display">\[
S(f) =
\cos(\phi)Absorption(f)+
\sin(\phi)Dispersion(f)+
m(f-f_0)+b
\]</span></p>
</div>
<div id="water" class="section level2 hasAnchor">
<h2 class="hasAnchor">Water<a href="#water" class="anchor-section"
aria-label="Anchor link to header"></a></h2>
<p>The unsurpressed water signal is fitted with a Lorentzian-Gaussian
model with a linear baseline:</p>
<p><span class="math display">\[
S(f) =
\frac{\cos(\phi)A+\sin(\phi)A\gamma(f-f_0)}
{\gamma^2(f-f_0)^2+1}
\exp[\sigma(f-f_0)^2]+
m(f-f_0)+b
\]</span></p>
<p><img id="img_75" src="images/signal-modeling/water.png" alt="Illustration of the water model"></p>
<p><br></p>
<div id="references" class="section level3 unnumbered hasAnchor">
<h3 class="unnumbered hasAnchor">References<a href="#references"
class="anchor-section" aria-label="Anchor link to header"></a></h3>
<div id="refs" class="references csl-bib-body">
<div id="ref-Evans2013" class="csl-entry">
<div class="csl-left-margin">1. </div><div
class="csl-right-inline">Evans CJ, Puts NAJ, Robson SE, et al. <span
class="nocase">Subtraction artifacts and frequency (Mis-)alignment in
J-difference GABA editing</span>. <em>Journal of Magnetic Resonance
Imaging</em>. 2013;38(4):970-975. doi:<a
href="https://doi.org/10.1002/jmri.23923">10.1002/jmri.23923</a></div>
</div>
</div>
</div>
</div>
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